How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]
draw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));
draw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));
label("1",(3,8),SSE);
label("2",(7,8),SSW);
label("3",(10,4),2W);
label("4",(9,0),NW+NNW);
label("5",(1,0),NE+NNE);
label("6",(0,4),2E);
[/asy]
Solution: The angles numbered 1, 3, and 5 are the three interior angles of a triangle, so they sum to $180^\circ$.  Similarly, the angles numbered 2, 4, and 6 are the three interior angles of a triangle, so they also sum to $180^\circ$.  Combining these, the sum of all six angle measures is $180^\circ + 180^\circ = \boxed{360^\circ}$.